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Hindman's theorem, ultrafilters, and reverse mathematics

Published online by Cambridge University Press:  12 March 2014

Jeffry L. Hirst
Jeffry L. Hirst*
Affiliation:
Department of Mathematical Sciences, Appalachian State University, Boone, North Carolina 28608, USA, E-mail: jlh@math.appstate.edu, URL: www.mathsci.appstate.edu/~jlh
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Abstract

Assuming CH. Hindman [2] showed that the existence of certain ultrafilters on the power set of the natural numbers is equivalent to Hindman's Theorem. Adapting this work to a countable setting formalized in RCA0, this article proves the equivalence of the existence of certain ultrafilters on countable Boolean algebras and an iterated form of Hindman's Theorem, which is closely related to Milliken's Theorem. A computable restriction of Hindman's Theorem follows as a corollary.

Keywords

Hindman's TheoremMilliken's Theoremreverse mathematicscomputability
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 1 , March 2004 , pp. 65 - 72
Copyright
Copyright © Association for Symbolic Logic 2004

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References

REFERENCES

[1]Blass, Andreas R., Hirst, Jeffry L., and Simpson, Stephen G., Logical analysis of some theorems of combinatorics and topological dynamics, Logic and combinatorics (Areata, Calif., 1985), Contemporary Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1987, pp. 125–156.CrossRefGoogle Scholar
[2]Hindman, Neil, The existence of certain ultra-filters on N and a conjecture of Graham and Rothschild, Proceedings of the American Mathematical Society, vol. 36 (1972), pp. 341–346.Google Scholar
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